Introduction to Quantum Mechanics · Problem A.25

Afterword — David J. Griffiths

Physics — Quantum Mechanics · eigenvectors-eigenvalues

Problem

Consider the matrices

\mathsf{A} = \begin{pmatrix} 1 & 4 & 1 \\ 4 & -2 & 4 \\ 1 & 4 & 1 \end{pmatrix}, \quad \mathsf{B} = \begin{pmatrix} 1 & -2 & -1 \\ -2 & 2 & -2 \\ -1 & -2 & 1 \end{pmatrix}.

(a) Verify that they are diagonalizable and that they commute. (b) Find the eigenvalues and eigenvectors of

\mathsf{A}

and verify that its spectrum is nondegenerate. (c) Show that the eigenvectors of

\mathsf{A}

are eigenvectors of

\mathsf{B}

as well.